Moore plane

Definition

• Elements of the local basis at points with are the open discs in the plane which are small enough to lie within.
• Elements of the local basis at points are sets where A is an open disc in the upper half-plane which is tangent to the x axis at p.

Properties

• The Moore plane is separable, that is, it has a countable dense subset.
• The Moore plane is a completely regular Hausdorff space, which is not normal.
• The subspace of has, as its subspace topology, the discrete topology. Thus, the Moore plane shows that a subspace of a separable space need not be separable.
• The Moore plane is first countable, but not second countable or Lindelöf.
• The Moore plane is not locally compact.
• The Moore plane is countably metacompact but not metacompact.

  Proof that the Moore plane is not normal

1. On the one hand, the countable set of points with rational coordinates is dense in M; hence every continuous function is determined by its restriction to, so there can be at most many continuous real-valued functions on M.
2. On the other hand, the real line is a closed discrete subspace of M with many points. So there are many continuous functions from L to. Not all these functions can be extended to continuous functions on M.
3. Hence M is not normal, because by the Tietze extension theorem all continuous functions defined on a closed subspace of a normal space can be extended to a continuous function on the whole space.